Stokes Flow Limit Research Articles

From an obliquely falling rod in a viscous fluid to the motion of suspended magnetic bead chains that are driven by a gradient magnetic field and that make an arbitrary angle with the magnetic force vector: A Stokes flow study.

In view of the growing role of magnetic particles under magnetic field influence in medical and other applications, and perforce the bead chaining, it is important to understand more generally the chain dynamics. As is well known, in the presence of a magnetic field, magnetic beads tend to form chains that are aligned with the magnetic field vector. In addition, if there is a magnetic field gradient, there will be a magnetic force acting on this chain. The main goal of the present research is to study the motion of a magnetic bead chain that makes an arbitrary angle with the magnetic force vector in the Stokes flow limit, that is, in the limit of zero Reynolds number. We used the public-domain computer program HYDRO++ to calculate the mobility matrix, which relates the magnetic force acting on the chain to the velocity of the chain, for a chain of N beads making an arbitrary angle with the magnetic force vector. Because of the presence of off-diagonal elements of the mobility matrix, as the chain is drawn in the direction of the magnetic force, it is also deflected to the side. We derived analytic solutions for this motion. Also, for bead chains moving in directions both parallel and perpendicular to their lengths, we fit three-parameter functions to solutions from HYDRO++. We found the fits to be excellent. Combining these results with the analytic solutions, we obtained expressions for the velocity components for the bead chains that provide excellent fits to HYDRO++ solutions for arbitrary angles. Finally, we apply the methodology used for the bead chain studies to the study of an obliquely falling rod in a viscous fluid and derive analytic solutions for the velocity components of the obliquely falling rod.

Translation-deformation coupling effects on the Rayleigh instability of an electrodynamically levitated charged droplet.

The breakup pathway of the Rayleigh fission process observed in the past experiments carried out using high-speed imaging of a charged drop levitated in an AC quadrupole trap has shown to exhibit several cycles of shape and center-of-mass oscillations followed by asymmetric breakup by ejecting a jet in the upward direction (i.e., opposite to the direction of gravity). We recently attempted to explain this using boundary integral simulations in the Stokes flow limit, wherein the position of the droplet and the polarity of the end cap electrodes were assigned using physical arguments, and the center-of-mass motion was not estimated consistently invoking quasi-static conditions. In this work, we explain the experimental observation of upward breakup of charged droplets in a quadrupolar field, using numerical calculations based on the boundary element method considering inviscid droplets levitated electrodynamically using quadrupole electric fields. The center-of-mass motion and the end cap are consistently calculated in the numerical scheme. The simulations show that the gravity-induced downward shift in the equilibrium position of the drop in the trap causes significant, large-amplitude shape oscillations superimposed over the center-of-mass oscillations of the drop. An important observation here is that the shape oscillations due to the applied quadrupole fields result in sufficient deformations that act as triggers for the onset of the instability below the Rayleigh limit, thereby admitting a subcritical instability. The center-of-mass oscillations of the droplet within the trap, which follow the applied frequency, are out of phase with the applied AC signal. Thus the combined effect of shape deformations and dynamic position of the drop leads to an asymmetric breakup such that the Rayleigh fission occurs upward via the ejection of a jet at the north pole of the deformed drop.

Impact of surface viscosity on the stability of a droplet translating through a stagnant fluid

This study examines the impact of interfacial viscosity on the stability of an initially deformed droplet translating through an unbounded quiescent fluid. The boundary-integral formulation is employed to investigate the time evolution of a droplet in the Stokes flow limit. The droplet interface is modelled using the Boussinesq–Scriven constitutive relationship having surface shear viscosity $\eta _\mu$ and surface dilatational viscosity $\eta _\kappa$ . We observe that, below a critical value of the capillary number, $Ca_C$ , the initially perturbed droplet reverts to its spherical shape. Above $Ca_C$ , the translating droplet deforms continuously, growing a tail at the rear end for initial prolate perturbations and a cavity for initial oblate perturbations. We find that surface shear viscosity inhibits the tail/cavity growth at the droplet's rear end and increases the $Ca_C$ compared with a clean droplet. In contrast, surface dilatational viscosity increases tail/cavity growth and lowers $Ca_C$ compared with a clean droplet. Surprisingly, both shear and dilatational surface viscosity appear to delay the time at which pinch off occurs, and hence satellite droplets form. Lastly, we explore the combined influence of surface viscosity and surfactant transport on droplet stability by assuming a linear dependence of surface tension on surfactant concentration and exponential dependence of interfacial viscosities on the surface pressure. We find that pressure-thinning/thickening effects significantly affect the droplet dynamics for surface shear viscosity but play a small role for surface dilatational viscosity. We lastly provide phase diagrams for the critical capillary number for different values of the droplet's viscosity ratio and initial Taylor deformation parameter.

Linear stability of a thin fluid film interacting with its surrounding bulk fluid

Thin films can become unstable when attractive van der Waals forces overcome the stabilizing effects of surface tension and viscous forces. In many applications, the effect of the surrounding bulk fluid cannot be neglected when considering a thin film subject to perturbations. In this work, we examine the two limits of potential flow and Stokes flow in the surrounding bulks to derive dispersion relations in each limit. We show that the effect of the surrounding bulks cannot be ignored for many film–bulk fluid pairings and film thicknesses and present conditions for the validity of each regime. In particular, the potential-flow regime exists when van der Waals forces are sufficiently strong, while the Stokes-flow regime exists when the bulk dynamic viscosity is sufficiently large. Due to the nature of the dispersion relation in the Stokes-flow limit, several distinct scenarios are identified in the corresponding stability diagram, each involving the interplay of different forces. For example, a novel instability regime involving capillary–viscous interactions is identified for large film thicknesses. Finally, by enlisting multiple realistic fluid pairings and film thicknesses wherein such instabilities can occur, we demonstrate the practical relevance of our theoretical findings. This work enables the extension of thin film stability theory to the analysis of antibubbles, splashing molten metals and ionic liquids, Mesler entrainment of microbubbles in breaking waves, and emulsion stability.

Electrohydrodynamic-induced interactions between droplets

Dispersion of droplets in an emulsion is commonly seen in several chemical, pharmaceutical and petroleum industries. Electric field has been shown to affect the stability of these dispersions. We study the dynamics of a pair of leaky dielectric droplets in a leaky dielectric liquid in the presence of an externally applied electric field. A pair of droplets may coalesce or repel each other in the presence of an electric field. Interactions between a pair of drops have been shown to be governed by the ratio $\varepsilon _r/\sigma _r$ , where $\varepsilon _r$ and $\sigma _r$ are the ratios of drop to ambient fluid electric permittivities and conductivities, respectively. When inertia is neglected, the droplets approach each other if $\varepsilon _r/\sigma _r > 1$ , whereas droplets repel when $\varepsilon _r/\sigma _r < 1$ . However, inclusion of inertia permits interesting transient behaviour, where the droplets may attract due to the electrostatic dipole–dipole attraction even for $\varepsilon _r/\sigma _r < 1$ . The approach velocity then is governed by the electrostatic forces and varies as $1/h^4$ , where $h$ is the separation distance between the droplets, in contrast to being hydrodynamically driven as predicted in the Stokes flow limit by Baygents et al. (J. Fluid Mech., vol. 368, 1998, pp. 359–375). For compound droplets, interactions between droplets are essentially governed by the electrical properties of the outer droplet and the ambient fluid. However, transient dynamics may also result in the breakup of a compound droplet and lead to formation of single droplets.

Ferro-hydrodynamic interactions between ferrofluid droplet pairs in simple shear flows

A numerical investigation on the interactions between a pair of equal-sized ferrofluid droplets in a simple shear flow subjected to uniform magnetic fields is carried out in this paper. Our numerical model utilizes a finite element method based level set algorithm to trace the topological changes along the droplet interface in two phase flows by coupling the flow and magnetic fields. Systematic numerical simulations with a well-resolved droplet interface are carried out in the Stokes flow limit (i.e., Re=0.03) to explore the effects of magnetic field direction, strength, and initial vertical offset on the pairwise interactions between the droplets. The findings indicate that in a solitary shear flow, a critical capillary number Cacr exists, beyond which the droplets exhibit separation through a sliding-over motion, instead of coalescence. Applying a uniform magnetic field along α=0° (i.e., parallel to the flow direction) results in a faster coalescence between the droplets. At α=90°, a critical magnetic bond number Bocr appears where the droplets show a reversing motion behavior, instead of a passing-over motion leading to coalescence. In contrast, at α=45°, the droplets separate away from each other, following a reversal motion at a capillary number where the droplets usually coalesce with each other in a solitary shear flow. Moreover, the effect of initial vertical offset on the interaction behavior of the droplets is examined, and it is found that the collision event is dependent on the initial vertical separation distance between the droplets. Increasing the vertical separation distance along α=0° initiates faster coalescence, while at α=90°, coalescence is delayed. Furthermore, at α=45°, the droplets undergo a reversing motion and experience larger migration at increased vertical separation distances.

Effect of the Quadrupolar Trap Potential on the Rayleigh Instability and Breakup of a Levitated Charged Droplet.

The experimental demonstration of Rayleigh instability that results in the breakup of a charged droplet, levitated in a quadrupole trap, has been investigated in the literature, but only scarcely. We report here the asymmetric breakup of a charged drop, levitated in a loose trap, wherein the droplet is stabilized at an off-center location in the trap. This aspect of levitation leads to an asymmetric breakup of the charged drop, predominantly in a direction opposite to that of gravity. In the present work, we report the evidence of successive events of the deformation and breakup of a charged drop and its subsequent relaxation after jet ejection using high-speed imaging at a couple of hundred thousand frames per second. Several relevant aspects of this phenomenon such as the effect of the electrodynamic (ED) trap parameters in terms of the applied potential as well as physical parameters such as the size of the drop, gravity, and conductivity on the characteristics of droplet breakup are explored. A clear effect of the trap strength on the deformation (both symmetric and asymmetric) is observed. Moreover, the cone angle at the pole undergoing asymmetric breakup is almost independent of the applied field investigated in the experiments. All of the experimental observations are compared with numerical simulations carried out using the boundary element method (BEM) in the Stokes flow limit. The BEM simulations are also extended to other experimentally achievable parameters. It is observed that the breakup in our study is mostly field-influenced and not field-induced. A plausible theory for the observations is reported, and a sensitive role of the sign of the charge on the droplet and the sign of the end-cap potential, as well as the off-center location of the droplet in the trap, is elucidated.

Large-deformation electrohydrodynamics of an elastic capsule in a DC electric field

The dynamics of a spherical elastic capsule, containing a Newtonian fluid bounded by an elastic membrane and immersed in another Newtonian fluid, in a uniform DC electric field is investigated. Discontinuity of electrical properties, such as the conductivities of the internal and external fluid media as well as the capacitance and conductance of the membrane, leads to a net interfacial Maxwell stress which can cause the deformation of such an elastic capsule. We investigate this problem considering well-established membrane laws for a thin elastic membrane, with fully resolved hydrodynamics in the Stokes flow limit, and describe the electrostatics using the capacitor model. In the limit of small deformation, the analytical theory predicts the dynamics fairly satisfactorily. Large deformations at high capillary number, though, necessitate a numerical approach (axisymmetric boundary element method in the present case) to solve this highly nonlinear problem. Akin to vesicles, at intermediate times, highly nonlinear biconcave shapes along with squaring and hexagon-like shapes are observed when the outer medium is more conducting. The study identifies the essentiality of parameters such as high membrane capacitance, low membrane conductance, low hydrodynamic time scales and high capillary number (the ratio of the destabilizing electric force to the stabilizing elastic force) for observation of these shape transitions. The transition is due to large compressive Maxwell stress at the poles at intermediate times. Thus such shape transition can be seen in spherical globules admitting electrical capacitance, possibly irrespective of the nature of the interfacial restoring force.

Effect of nonuniform electric field on the electrohydrodynamic motion of a drop in Poiseuille flow

The effect of a nonuniform electric field on the electrohydrodynamic motion of a leaky dielectric suspended drop in the presence of background Poiseuille flow is investigated analytically. Considering the nonuniform electric field to be a linear combination of uniform and quadrupole fields, the velocity of a force-free drop positioned at the flow centerline is obtained. The drop velocity is strongly influenced by the surface charge distribution and drop shape. In the Stokes flow limit, we employ an asymptotic method considering weak surface charge convection and small shape deformation. The present study shows the importance of type of nonuniform electric field (converging or diverging in the direction of the Poiseuille flow), strength of the electric field relative to the Poiseuille flow, and material property ratios on the magnitude and direction of drop motion in the presence of flow curvature. In the presence of a nonuniform electric field, the flow curvature can increase or decrease the drop velocity as compared with a uniform flow case. The converging electric field always drives a perfectly conducting drop in the direction of the Poiseuille flow with increased velocity, while the diverging electric field can drive the drop in either direction, depending on the relative strength of the applied electric field. Shape deformation increases the velocity of a perfectly conducting drop in the converging electric field, while shape deformation increases/decreases the velocity magnitude of a perfectly conducting drop in the diverging electric field. The converging electric field always drives a perfectly dielectric drop in the direction of the Poiseuille flow with increased (or decreased) velocity when the drop phase permittivity is greater (or less) than the medium phase permittivity. The diverging electric field can move a perfectly dielectric drop in either direction, depending on the strength of the electric field relative to the Poiseuille flow and drop-to-medium permittivity ratio. Shape deformation increases the velocity magnitude of a perfectly dielectric drop for larger permittivity ratios. For leaky dielectric drops, both surface convection and shape deformation can increase or decrease the drop velocity in nonuniform electric field, depending on the electrohydrodynamic properties of the drop and the suspending medium.

Hydrodynamic force on a sphere normal to an obstacle due to a non-uniform flow

For a small sphere suspended in a background fluid flow near an obstacle, we calculate the hydrodynamic force on the sphere in the direction normal to the boundary of the obstacle. Using the Lorentz reciprocal theorem, we obtain analytical expressions for the normal force in the Stokes flow limit, valid for arbitrary separations of the particle from the obstacle, both for solid obstacles and those with free surfaces. The main effect of the boundary is to produce a normal force proportional to extensional flow gradients in the vicinity of the particle. The strength of this force is greatest when the separation between the surfaces of the particle and the obstacle is small relative to the particle size. While the magnitude of the force weakens for large separations between the sphere and the obstacle (decaying quadratically with separation distance), it can significantly modify Faxén’s law even at modestly large separation distances. In addition, we find a second force contribution due to the curvature of the background flow normal to the obstacle, which is also important when the sphere is close to the obstacle. The results of the theory are of importance to the dynamics of particles in confined geometries, whether bounded by a solid obstacle, the wall of a channel or a gas bubble.

A complex-valued first integral of Navier-Stokes equations: Unsteady Couette flow in a corrugated channel system

For a two-dimensional incompressible viscous flow, a first integral of the governing equations of motion is constructed based on a reformulation of the unsteady Navier-Stokes equations in terms of complex variables and the subsequent introduction of a complex potential field; complementary solid and free surface boundary conditions are formulated. The methodology is used to solve the challenging problem of unsteady Couette flow between two sinusoidally varying corrugated rigid surfaces utilising two modelling approaches to highlight the versatility of the first integral. In the Stokes flow limit, the results obtained in the case of steady flow are found to be in excellent agreement with corresponding investigations in the open literature. Similarly, for unsteady flow, the results are in accord with related investigations, exploring material transfer between trapped eddies and the associated bulk flow, and vice versa. It is shown how the work relates to the classical complex variable method for solving the biharmonic problem and perspectives are provided as to how the first integral may be further utilised to investigate other fluid flow features.

The effect of surface charge convection and shape deformation on the settling velocity of drops in nonuniform electric field

The electrohydrodynamic settling of a leaky dielectric drop suspended in another leaky dielectric medium of unbounded extent in the combined presence of gravity and a nonuniform (combination of uniform and quadrupole) electric field is investigated theoretically in the Stokes flow limit. The present study incorporates both the effects of surface charge convection and shape deformation on the drop settling speed. The drop settling speed in the presence of an electric field is governed by three dimensionless groups: (i) capillary number Ca (the ratio of viscous to capillary stresses), (ii) electric Reynolds number ReE (the ratio of charge relaxation to convection time scales), and (iii) Masson number M (the ratio of electric to viscous stresses). Depending on the material properties of the drop and suspending medium, the strength of the applied electric field, and the drop radius, the following two different kinds of physical systems are identified for which asymptotic solutions for the settling velocity is obtained: (i) weak charge convection and small shape deformation (i.e., ReE≪1, Ca≪1) but with appreciable electric field strength (M∼1) and (ii) weak electric field (i.e., M≪1) but with appreciable charge convection ReE∼1. For a perfectly conducting drop suspended in a leaky (or perfectly) dielectric medium, the application of a uniform and/or quadrupole electric field always deforms the drop into a prolate shape which leads to an increase in drop settling speed. For a perfectly dielectric drop suspended in a perfectly dielectric medium, the application of a quadrupole electric field can deform the drop into either a prolate or an oblate shape which leads to an increase or decrease in the drop speed depending on the drop to medium permittivity ratio. Combined presence of uniform and quadrupole fields induces a dielectrophoretic force on the drop which not only alters the drop speed but also can change the direction of drop motion depending on the values of physical parameters. The present study establishes that surface charge convection plays an important role in determining the drop settling speed for a leaky dielectric drop suspended in a leaky dielectric medium. It is shown that the charge-convection-induced correction in the drop settling speed is larger in magnitude in the presence of a quadrupole electric field as compared to a uniform electric field; this is due to a more intense circulation inside the drop in the presence of a quadrupole electric field.

On the scaling of the slip velocity in turbulent flows over superhydrophobic surfaces

Superhydrophobic surfaces can significantly reduce hydrodynamic skin drag by accommodating large slip velocity near the surface due to entrapment of air bubbles within their micro-scale roughness elements. While there are many Stokes flow solutions for flows near superhydrophobic surfaces that describe the relation between effective slip length and surface geometry, such relations are not fully known in the turbulent flow limit. In this work, we present a phenomenological model for the kinematics of flow near a superhydrophobic surface with periodic post-patterns at high Reynolds numbers. The model predicts an inverse square root scaling with solid fraction, and a cube root scaling of the slip length with pattern size, which is different from the reported scaling in the Stokes flow limit. A mixed model is then proposed that recovers both Stokes flow solution and the presented scaling, respectively, in the small and large texture size limits. This model is validated using direct numerical simulations of turbulent flows over superhydrophobic posts over a wide range of texture sizes from L+ ≈ 6 to 310 and solid fractions from ϕs = 1/9 to 1/64. Our report also embarks on the extension of friction laws of turbulent wall-bounded flows to superhydrophobic surfaces. To this end, we present a review of a simplified model for the mean velocity profile, which we call the shifted-turbulent boundary layer model, and address two previous shortcomings regarding the closure and accuracy of this model. Furthermore, we address the process of homogenization of the texture effect to an effective slip length by investigating correlations between slip velocity and shear over pattern-averaged data for streamwise and spanwise directions. For L+ of up to O(10), shear stress and slip velocity are perfectly correlated and well described by a homogenized slip length consistent with Stokes flow solutions. In contrast, in the limit of large L+, the pattern-averaged shear stress and slip velocity become uncorrelated and thus the homogenized boundary condition is unable to capture the bulk behavior of the patterned surface.

Non-isothermal bubble rise: non-monotonic dependence of surface tension on temperature

AbstractWe study the motion of a bubble driven by buoyancy and thermocapillarity in a tube with a non-uniformly heated walls, containing a so-called ‘self-rewetting fluid’; the surface tension of the latter exhibits a parabolic dependence on temperature, with a well-defined minimum. In the Stokes flow limit, we derive the conditions under which a spherical bubble can come to rest in a self-rewetting fluid whose temperature varies linearly in the vertical direction, and demonstrate that this is possible for both positive and negative temperature gradients. This is in contrast to the case of simple fluids whose surface tension decreases linearly with temperature, for which bubble motion is arrested only for negative temperature gradients. In the case of self-rewetting fluids, we propose an analytical expression for the position of bubble arrestment as a function of other dimensionless numbers. We also perform direct numerical simulation of axisymmetric bubble motion in a fluid whose temperature increases linearly with vertical distance from the bottom of the tube; this is done for a range of Bond and Galileo numbers, as well as for various parameters that govern the functional dependence of surface tension on temperature. We demonstrate that bubble motion can be reversed and then arrested only in self-rewetting fluids, and not in linear fluids, for sufficiently small Bond numbers. We also demonstrate that considerable bubble elongation is possible under significant wall confinement, and for strongly self-rewetting fluids and large Bond numbers. The mechanisms underlying the phenomena observed are elucidated by considering how the surface tension dependence on temperature affects the thermocapillary stresses in the flow.

Gravity and capillary pressure-driven drainage in a vertical Hele-Shaw cell: Thin film deposition

The stable-downward vertical displacement of an air–liquid meniscus in a two-dimensional Hele-Shaw cell is studied in the limit of small capillary (Ca⁎) and Bond (BoH⁎) numbers. Subsequently, a thin film of thickness h is deposited on the substrate walls for Ca⁎⪯¡BoH⁎. Using a reverse Washburn analysis as it applies to two-dimensional geometries in the Stokes flow limit, we develop mathematical expressions for the speed of the average location of the meniscus x¯N spanning the smallest gap dimension a. Furthermore, we show that the other spanwise meniscus is stable using classic linear stability analysis for immiscible fluid displacements. Asymptotic analysis of the average meniscus speed yields an expression for the film thickness as a function of several measurable parameters. Experiments are performed using silicone oil (kinematic viscosities 1×10−5, 1×10−4 and 1×10−3 m2/s) while varying the initial displacement height of the meniscus before descent x¯N0, and the Hele-Shaw cell gap spacing. Analysis of the transient film thickness h, estimated from the average meniscus position, speed and physical properties, is in good agreement and suggests self-similarity.

Coalescence of surfactant covered drops in extensional flows: Effects of the interfacial diffusivity

Boundary integral simulations and scaling theory were employed to study the effects of insoluble surfactant surface diffusivity Ds and concentration Γ on the coalescence process of two equal-sized viscous drops. The drops underwent head-on collisions in a biaxial extensional flow, in the Stokes flow limit and low capillary numbers. The simulations were compared with the drainage time experiments of Yoon et al. [Phys. Fluids 19, 023102 (2007)10.1063/1.2409735] concerned with a polymeric system, polybutadiene (PBd) drops in a polydimethylsyloxane (PDMS) matrix, stabilized by block-copolymers acting as insoluble surfactants to explain the mechanism underneath their findings. An ad hoc equation of state, derived by mean field theory, specific for the block-copolymers in the experiments of Yoon et al., able to match the experimental surface tension data without fitting parameters, was used. We were able to reproduce the experimental drainage time data, although an additional attractive force, besides the usual van der Waals interactions, had to be introduced for high block-copolymer concentrations, probably as a result of the entropic attraction between the two facing dry brushes formed in the thin film between the two drops. According to simulations, the puzzling experimental drainage time transition for low surfactant concentrations, from high drainage time to low drainage time as Ca increases, was a consequence of the oscillating behavior of the minimum film thickness, which takes place for Marangoni numbers Ma < 5 and surface Peclet number Pes > 60. In this regard, a master curve was obtained for the scaled relative minimum film thickness attained during the oscillation as a function of Ma. This enabled to determine both the minimum value of the dimensionless attractive forces to avoid coalescence for each concentration studied and the range of Ma that favors early coalescence. The coalescence process was found very sensitive to Pes and for Pes O(100–1000) even trace amounts of surfactants can be as effective stabilizers as high surfactant concentrations. Moreover, for the polymeric system of interest, the range of Ds in which the drainage time changes from the saturation value to the clean interface value was computed as a function of the surfactant concentration. In the specific, for the PDMS/PBd system of interest the Ds range studied was O(10−12–10−5 cm2 s−1). Additionally, our scaling analysis further validates our simulations, also highlighting the effect on the drainage process of the different parameters, in particular, of the external pushing force, which is increased compared to a clean interface system, as Pes is increased or as the surfactant concentration is increased, because of the reduction in the interfacial mobility of the drop. Finally, our study suggests that matching simulations with four-roll mill drainage time experiments can be an effective method to determine block-copolymer surface diffusivity.

Relaxation and coalescence of two equal-sized viscous drops in a quiescent matrix

AbstractHead-on collisions of two equal-sized viscous drops in a biaxial extensional flow were simulated using the boundary integral method in the Stokes flow limit, for capillary numbers of the order of $1{0}^{\ensuremath{-} 4} $–$1{0}^{\ensuremath{-} 1} $, typical of flow-induced coalescence experiments. At a certain point in time, during the drainage process, the flow was abruptly stopped and the time-dependent dynamics of drop deformation (relaxation) was followed to discern whether the pair of drops would eventually coalesce. The concept of coalescence probability was used to study the evolution of probable collisions. The polymeric system of polybutadiene (PBd) drops undergoing head-on collisions in a polydimethylsiloxane (PDMS) matrix, previously well-characterized both experimentally and numerically by Yoon et al. (Phys. Fluid, vol. 19, 2007, 102102), in which both fluids were Newtonian under the experimental conditions, was used as our reference. Film shapes, velocity profiles and pressure distributions were studied for initially parabolic or dimpled thin film shapes. It was shown that micrometre-sized drops undergoing relaxation can coalesce in the capillary number range studied, which also included cases of hindered coalescence and cases in which the flow interaction time for the collision was smaller than the drainage time; thus, this phenomenon could influence the final drop size distribution of blends. Further, these findings could be of interest in interpreting stop–strain experiments, in the case of a sudden change in flow conditions and in population balance studies of drops in blends.

Hydrodynamic irreversibility in particle suspensions with nonuniform strain

A dynamical phase transition from reversible to irreversible behavior occurs when particle suspensions are subjected to uniform oscillatory shear, even in the Stokes flow limit. We consider a more general situation with nonuniform strain (e.g., oscillatory channel flow), which is observed to exhibit markedly different dynamics. The onset of irreversibility is delayed, and occurs simultaneously across the entire channel. This behavior is only partially explained by self-organization and shear-induced migration. The onset is accompanied by long-range correlated particle motion even at the channel center, where the strain is negligible; this motion prevents the system from evolving into a reversible state.

Influence of surfactant solubility on the deformation and breakup of a bubble or capillary jet in a viscous fluid

In a previous study [M. Hameed et al., J. Fluid Mech. 594, 307 (2008)] the authors investigated the influence of insoluble surfactant on the evolution of a stretched, inviscid bubble surrounded by a viscous fluid via direct numerical simulation of the Navier–Stokes equations, and showed that the presence of surfactant can cause the bubble to contract and form a quasisteady slender thread connecting parent bubbles, instead of proceeding directly toward pinch-off as occurs for a surfactant-free bubble. Insoluble surfactant significantly retards pinch-off and the thread is stabilized by a balance between internal pressure and reduced capillary pressure due to a high concentration of surfactant that develops during the initial stage of contraction. In the present study we investigate the influence of surfactant solubility on thread formation. The adsorption-desorption kinetics for solubility is in the diffusion controlled regime. A long-wave model for the evolution of a capillary jet is also studied in the Stokes flow limit, and shows dynamics that are similar to those of the evolving bubble. With soluble surfactant, depending on parameter values, a slender thread forms but can pinch-off later due to exchange of surfactant between the interface and exterior bulk flow.

Eddy genesis and manipulation in plane laminar shear flow

Eddy formation and presence in a plane laminar shear flow configuration consisting of two infinitely long plates orientated parallel to each other is investigated theoretically. The upper plate, which is planar, drives the flow; the lower one has a sinusoidal profile and is fixed. The governing equations are solved via a full finite element formulation for the general case and semianalytically at the Stokes flow limit. The effects of varying geometry (involving changes in the mean plate separation or the amplitude and wavelength of the lower plate) and inertia are explored separately. For Stokes flow and varying geometry, excellent agreement between the two methods of solution is found. Of particular interest with regard to the flow structure is the importance of the clearance that exists between the upper plate and the tops of the corrugations forming the lower one. When the clearance is large, an eddy is only present at sufficiently large amplitudes or small wavelengths. However, as the plate clearance is reduced, a critical value is found, which triggers the formation of an eddy in an otherwise fully attached flow for any finite amplitude and arbitrarily large wavelength. This is a precursor to the primary eddy to be expected in the lid-driven cavity flow, which is formed in the limit of zero clearance between the plates. The influence of the flow driving mechanism is assessed by comparison with corresponding solutions for the case of gravity-driven fluid films flowing over an undulating substrate. When inertia is present, the flow generally becomes asymmetrical. However, it is found that for large mean plate separations the flow local to the lower plate becomes effectively decoupled from the inertia dominated overlying flow if the wavelength of the lower plate is sufficiently small. In such cases the local flow retains its symmetry. A local Reynolds number based on the wavelength is shown to be useful in characterizing these large-gap flows. As the mean plate separation is reduced, the form of the asymmetry caused by inertia changes and becomes strongly dependent on the plate separation. For lower plate wavelengths which do not exhibit a kinematically induced secondary eddy, an inertially induced secondary eddy can be created if the mean plate separation is sufficiently small and the global Reynolds number is sufficiently large.